The zener is there to allow a faster decay of the current during switch off. If you put a diode across the coil then turn on the switch current starts to flow and increases linearly with time. When the switch turns off the inductor tends to want to keep the current flowing, and if the diode is there it flows until it is dissipated in the diode drop and the coil resistance. With the zener diode the same power is now dissipated in a much higher resistance device than the coil so it decays faster. You can use a regular resistor and parallel capacitor there in place of a zener with a similar effect but with a trade off of the peak voltage across the switch being proportional to the pulse width, which means you need a switching device with a maximum voltage of at least 2x the supply rail and the diode has to have a pulse rating equal to the peak current in the coil. The zener gives a more closely defined peak voltage and a lower peak current through the diode. You also do not need to have as fast a switching diode there.
Another method is to use a zener diode across the switch, selected so as to not exceed the max rated off state voltage of the switch, though this device has to be a fast turn on transient suppression diode capable of handling the power in the coil.